The methods according to exemplary embodiments can be illustrated by taking for the sake of pedagogy, applications pertaining to spatial filtering, a field known as array processing. In this context, the Hermetic Transform, as described in U.S. Pat. No. 8,064,408 (which is incorporated by reference as if fully set forth herein), can be utilized in array processing as a substitute for what can be referred to as “Super-Gain” or “Super-Directive” array processing. A goal of such processing is to overcome the classical diffraction limit, wherein angular resolution (“beam-width”) is known to be approximately given by the following expression:δθ˜λ/D  (1)with δθ being the angular resolution (“beam-width”) in radians, λ the wavelength of the signal arrival at the array, and D the array characteristic dimension. For example, the above formula generally describes the angular resolution of antenna array of dimension D (assumed to be at least one wavelength in extent) to a radio wave of wavelength λ impinging on it, with the antenna oriented (or in the case of a multi-element phased-array, electronically “steered”) to have a maximum response in the direction of the source of the radio energy. The diffraction limit on resolution owes to spatial response of the array to signal arrivals from various directions which is characterized by a function known as the “diffraction pattern”, or “antenna pattern”. The antenna pattern corresponds to the spatial Fourier Transform of the array of element excitations at the individual antenna elements. The diffraction pattern shape is normally comprised of a “main lobe” (“beam”) which is steered in some particular direction, and a number of ancillary “side lobes”, which respond to energy arriving from directions away from the main lobe of the pattern.
Goals for antenna pattern design typically include (1) making the beam-width as narrow as possible in the main-lobe direction, and (2) mitigating the array response to signals arriving from directions other than that of the main lobe. As a result it is common to “weight” the outputs of individual antenna elements with a “windowing function”, such as a Hanning function, to minimize side-lobe responses, at the expense of a modestly broadened pattern main-lobe. Another typical goal of antenna design is to create directive “gain” against radio-frequency noise which is often taken to have a statistical distribution that is isotropic in direction of arrival and uniformly random in terms of wave polarization. For this specific case, the spatial correlation of random noise between elements can be shown to be exactly zero for the case of half-wavelength inter-element separation. Half wavelength spacing also corresponds to the maximum spacing between elements that will produce unambiguous beams having no spatial aliasing (“grating lobes”), and produces as well, the largest useful linear array dimension for an array having N elements, yielding the narrowest diffraction-limited beam width.
Given all of the above assumptions, and representing the signal waveform as complex, it is shown that the maximum directive gain is accomplished by applying complex weights that are the complex conjugates of the arriving waves, the so-called “spatial matched-filter” or what is referred to henceforth as a “matched-filter transform” (MFT) since a set of such weights transforms the signal from element space, to angle/wave-vector space. Let Σ be a matrix having elements Σij (with i being the row dimension and j the column dimension) comprised of the complex antenna response of element i and direction of arrival j. The matrix Σ is by convention referred to as the “array manifold”. The array manifold can be calculated (in principle) or measured empirically for any physical array. A matrix form of the MFT (with weighting of array inputs) can be defined according to the following expression:z=ΣHWX  (2)where z is an output beam (complex time series), W is a diagonal matrix that contains real weights (e.g., the Hanning function from the previous example) used to control the pattern sidelobes and X is a complex time “snapshot” vector (set of synchronous time samples) output from the antenna. Each row in the matrix ΣH (e.g., the i-th row) spatially filters out signals mostly from one particular direction (the i-th manifold direction of arrival). For the case of an “ideal manifold”, i.e. the signals arriving from various directions have plane wavefronts and are monochromatic, having a single frequency, the Matched Filter Transform corresponds to a Fourier Transform as applied to the spatially sampled aperture, a form sometimes also termed a “Butler Matrix.”
For the case where inter-element (antenna) spacing is less than a half-wavelength, the above assumptions generally do not hold. The noise deriving from sources external to the array becomes spatially correlated (element to element), and the beam broadens relative to its nominal value for an array having half wavelength spacing. The Hermetic Transform responds to a need and desire to gain utility by producing high resolution beams with correspondingly high directional gain (gain against externally derived ambient noise) while utilizing small (even ultra-small, sub-wavelength size) multi-element arrays, that are implicitly oversampled spatially. As described in related art by Woodsum, a “decomposable” form of Hermetic Transform can be created by modifying the above expression, Equation (2), to allow the weight matrix W to be more generally non-diagonal and complex, imposing a criterion to allow determination of W through an optimization procedure. If we construct an input matrix X, which is the Array Manifold Matrix itself, is constructed, then there is a mathematical statement of the requirement for WΣHWΣ=I  (3)
The identity matrix corresponds to the discrete form of the spatial delta function, i.e., the criterion is to create a beam that is as close as possible in some optimal sense to that of a spatial delta function. This type of beam cannot be actually achieved in practice, but represents an “ideal” beam with maximum resolution and gain against isotropic noise. The above equation can be solved for W and hence for the Hermetic Transform (H) which is defined as the following:H=ΣHW  (4)One very general solution for W is given by:W=[ΣΣH]#Σ(I)ΣH[ΣΣH]#  (5)where the identity matrix is shown explicitly; other desired beam responses than I could be substituted. The # symbol indicates the Moore-Penrose pseudo-inverse in Gelb's notation. In practice the pseudo-inverse is often created using the Singular Value Decomposition (SVD). The above method of solving for W and therefore for H, is often chosen in practice because it makes use of the spatial covariance between elements, well known from adaptive beam-forming. The matrix W, being in general non-diagonal, is effectively a “metric” for a transformed, non-Euclidean complex signal space, in which beams that would not be orthogonal in original, untransformed Euclidean signal space can be orthogonalized, in a least-squares or minimum norm sense, often to within machine precision. There are various other means of creating Hermetic Transforms some of which will be further discussed below.
Summarizing the above discussion, one conventional method of spatial filtering known as beam-forming, involves multiplying each array element by a set of individual real weights, prior to applying an MFT matrix in order to form multiple beams. This process lowers side-lobes levels at the expense of broadening the main lobe of each beam. By contrast, the decomposable Hermetic Transform approach applies a particular, optimized weight matrix (W) to the array data, prior to applying the MFT, which has the effect of narrowing the beam main-lobe, without increasing side lobe levels. Side-lobe control can be exercised using a more general spatial filtering approach based on the Hermetic Transform. An assumption in this process is the spatial over-sampling of the array, i.e., the array elements are significantly closer together than ½ wavelength (significantly higher spatial sampling than spatial Nyquist). The matrix W is solved for in such a fashion as to make a beam in a particular “look direction” Ω0˜(θk, φk) that is as close as possible in a minimum norm sense to a delta function beam in angle space. δ(Ω-Ω0). The solution is called “decomposable” because the transform is composed of the product of two matrices, a MFT part (ΣH) and a weight matrix part W. Only the array manifold Σ is required to create the Hermetic Transform. With hindsight, the problem could have instead been formulated thus: find a matrix H such thatHΣ=I  (6)in a minimum norm sense. The solution in this case produces the result shown below.H=I{ΣH[ΣΣH]#}=ΣH[ΣΣH]#  (7)
This solution suggests an approach that resembles a solution known from adaptive array theory which termed the Minimum Variance Distortionless Response (MVDR) approach, except that the noise covariance in MVDR has been replaced with a covariance matrix type form derived from the array manifold. A modification of the above result is suggested by the following ansatz. For a weight vector corresponding to one row of the Hermetic Transform—i.e., to form a beam directed in the direction indicated by angles (θk,φk)—apply the weight vectorwH(θk,φk)=ΣH(θk,φk; ω){[DkΣ][DkΣ]H}#  (8)where Dk is an operator that zeroes out the column of the matrix Σ corresponding to the direction (θk,φk). Essentially, this procedure is like MVDR, except that the interfering “noise” which is being nulled out consists of all angles of arrival, with exclusion of the “look” direction (θk,φk). The intuitive explanation is that an MFT beam tries only to correlate (matched-filter to) an arriving signal from a particular look direction with the array manifold response vector associated with that direction, while the Hermetic Transform tries to do the same thing while trying to null out all responses away from the look direction (θk,φk). In practice, the weight vector produced by the above expression is normalized to unity gain in each particular look direction, for example the direction given by (θk,φk), with the following procedurewH(θk,φk)>>>wH(θk,φk)[wH(θk,φk)Σ(θk,φk; ω)]−1  (9)
The non-decomposable form has the advantage of being a row-by-row solution for the transform which does not involve inverting or finding the SVD of extremely large matrices; the decomposable form can potentially become computationally unwieldy under some practical conditions. In many cases, when beams are normalized correspondingly, the results for both approaches produce results that are nearly identical, in a numerical sense.
Creation of Spatial Filters Using Hermetic Transforms
Also described in related art, are more general filters, including spatial filters, that can be created using Hermetic Transforms. An elemental transform applies the following mathematical operations:F=H#ΛH  (10)
The complex filter matrix F is of dimension M×M (for an M-element array) and can be interpreted in terms of the beam-transform space operations. First, the Hermetic Transform H is applied to an input signal vector (time snapshot from the array) in order to transform the signal into the wave-vector (or beam) domain. The result is then multiplied by a diagonal matrix Λ, which applies weights to each “beam”. Finally, the pseudo-inverse of the Hermetic Transform (H#) is applied to move back to the spatial domain. If Λ is chosen as the identity matrix, the signal would remain unchanged by applying the filter matrix F. If Λ is instead chosen as a matrix with all but one non-zero elements, having a “one” on the p-th row diagonal element, the filter will project out of the signal all of the data except for that part of that signal that lies in the Hermetic Transform beam pointed at the p-th look direction. This type of filter transform is referred to as a “simple” or “elemental” spatial “pole” analogous to a pole in the frequency domain response of a time-series filter. Similarly, if for Λ a modified identity matrix is chosen that has one diagonal element in the p-th row zeroed out, the transform F will remove data from one beam (look) direction, making a null in that direction. This type of transform is referred to as a “simple” or an “elemental” spatial “zero”. By adding weighted cascade products of elemental transforms together, it is possible to make nearly arbitrary spatial filters that can be designed and optimized so as to approach a desired spatial response. A variety of methods can be used to develop a filter cascade from the elemental filter section, in order to achieve desired properties, for example the Genetic Algorithm approach.
Self-Noise Compensation
Extra resolution of the Hermetic Transform comes at the expense of what can potentially be problematic “white noise gain”. The use of spatial oversampling removes this problem with respect to background noise (the background noise acquires non-diagonal covariance); however, if the internal self-noise (with diagonal covariance) due to receiving array electronic noise is sufficiently large, another term is added to the transform expression. This term is a “noise conditioning” matrix K given byK=RΣΣ[RNN+RΣΣ]#  (11)where RNN is the internal self-noise covariance, and RΣΣ is a scaled manifold covariance (=c ΣΣH). The conditioned Hermetic Transform is in this case given by the following expression.H=ΣHWK  (12)
This procedure filters each signal arrival as corrupted by self-noise, with K, in such a fashion as to make the conditioned sum of signals plus internal noise as close as possible in a minimum norm sense, to the signal arrivals alone. In practice, a relevant signal to noise ratio can be assumed for scaling purposes, and since internal self-noise can be measured by removing stimulus to the array, and is effectively a stationary random process, the creation of a conditioning matrix becomes a one-time issue (not a real-time issue).
Application of noise conditioning is often unnecessary, the need for it being completely dependent on the specifics of the problem at hand. Measurement of internal, electronic noise is usually not difficult, and the above technique can be made robust to uncertainty in this parameter, as well.
The principles and method construction for Hermetic Transform used in frequency spectrum analysis are similar to those outlined above. The manifold matrix is replaced by a matrix of column vectors where each column vector is a complex sinusoidal signal computed at instants of time that are multiples of the sampling period Ts, where Ts=1/(sampling frequency), for a set of m angular frequencies (ωm which essentially correspond to the angles of arrival in the spatial version of the transform. The “manifold” element for row—n and column—m is given by:Σ(n,m)=exp[i ωmnTs]  (13)
Transforms and filters are formed using the precisely identical formulas as for the spatial transform, except for making use of the appropriate “manifold” for frequency spectrum analysis (Equation 13). Hermetic Transforms also have a time-domain form, wherein the manifold matrix E is constructed of Fourier or Hermetic (frequency) Transforms of time-shifted versions of a replica signal, and the transform developed has the effect of replacing time-domain replica correlation with a transform producing correlations with much higher than conventional time resolution (e.g. U.S. Patent Application Publication No. US20150145716A1, “Radar using Hermetic Transforms”)
Additional background is provided in related art U.S. Pat. Nos. 8,948,718; 8,064,408; 9,154,353; 8,559,456; and 9,154,214, as well as (1) “Optimized Hermetic Transform Beam-forming of Acoustic Arrays Via Cascaded Spatial Filter Arrangements Derived Using A Chimerical Evolutionary Genetic Algorithm” Harvey C. Woodsum and Christopher M. Woodsum, Proceedings, International Congress on Acoustics, ICA-13, June 2013—Montreal, Canada; and (2) “Optimization of Cascaded Hermetic Transform Processing Architectures via a Chimerical Hybrid Genetic Algorithm”, C. M. Woodsum and H. C. Woodsum, Proceedings of the Sixteenth International Conference on Cognitive and Neural Systems (ICCNS), Boston University, May 30-Jun. 1, 2012.